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Mirrors > Home > ILE Home > Th. List > r19.3rm | GIF version |
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Ref | Expression |
---|---|
r19.3rm.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
r19.3rm | ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2240 | . . 3 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
2 | 1 | cbvexv 1918 | . 2 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴) |
3 | eleq1 2240 | . . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
4 | 3 | cbvexv 1918 | . . 3 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴) |
5 | biimt 241 | . . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) | |
6 | df-ral 2460 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
7 | r19.3rm.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
8 | 7 | 19.23 1678 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
9 | 6, 8 | bitri 184 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
10 | 5, 9 | bitr4di 198 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
11 | 4, 10 | sylbi 121 | . 2 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
12 | 2, 11 | sylbir 135 | 1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 Ⅎwnf 1460 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-cleq 2170 df-clel 2173 df-ral 2460 |
This theorem is referenced by: r19.28m 3512 r19.3rmv 3513 r19.27m 3518 indstr 9579 |
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